Inequalities for the inverses of the polygamma functions

Abstract

We provide an elementary proof of the left-hand side of the following inequality and give a new upper bound for it. $$\begin{aligned} \bigg [\frac{n!}{x-(x^{-1/n}+\alpha )^{-n}}\bigg ]^{\frac{1}{n+1}}&<((-1)^{n-1}\psi ^{(n)})^{-1}(x) \\&<\bigg [\frac{n!}{x-(x^{-1/n}+\beta )^{-n}}\bigg ]^{\frac{1}{n+1}}, \end{aligned}$$ where $$\alpha =[(n-1)!]^{-1/n}$$ and $$\beta =[n!\zeta (n+1)]^{-1/n}$$ , which was proved in Batir (J Math Anal Appl 328:452–465, 2007), and we prove the following inequalities for the inverse of the digamma function $$\psi $$ . $$\begin{aligned} \frac{1}{\log (1+e^{-x})}<\psi ^{-1}(x)< e^{x}+\frac{1}{2}, \quad x\in \mathbb {R}. \end{aligned}$$ The proofs are based on nice applications of the mean value theorem for differentiation and elementary properties of the polygamma functions.